Hybrid vertical/horizontal axis wind turbine for deep-water offshore installations

ABSTRACT

A wind-driven power generating system with a hybrid wind turbine mounted on a floating platform that heels relative to horizontal in the presence of a prevailing wind. The hybrid turbine has a turbine rotor with at least two rotor blades, each mounted to a turbine shaft by at least one strut, and the system is configured so that the shaft forms a predetermined non-zero operating heel angle relative to vertical in the presence of a prevailing wind at a predetermined velocity. The blades and struts are airfoils with predetermined aerodynamic characteristics that generate lift forces with components in the direction of rotation around the shaft of the blades and struts at the operating heel angle to drive an electrical generator carried by the platform. The system can be designed to generate maximum power at the predetermined heel angle or essentially constant power over a range of heel angles.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. provisional application No.62/283,662, filed Sep. 8, 2015, the entire contents of which areincorporated herein by reference. This application includes subjectmatter related to that disclosed in U.S. application Ser. No.15/138,000, filed Apr. 25, 2016, by the herein applicant. The entirecontents of application Ser. No. 15/138,000, and provisional applicationno. 62/178,917, filed Apr. 23, 2015, from which it claims priority, arealso incorporated herein by reference.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates to floating lift-driven wind turbines, andmore particularly, to a floating hybrid vertical/horizontal axis windturbine designed to assume a predetermined heel angle that can be chosento maximize the electrical power generated by the wind at a specificoffshore location with known environmental conditions and to a floatinghybrid wind turbine that generates nearly constant power over a range ofheel angles.

Description of Related Art

The power industry is increasingly developing ways of generatingelectricity other than fossil fuels and nuclear energy. Many sources ofrenewable energy are being considered, but wind is one of the mostpopular. The number of “wind farms,” sometimes comprising hundreds ofwind turbines arrayed over a several square miles, is steadilyincreasing. However, the only type of wind turbine currently used forlarge, utility-scale power generation is the horizontal axis windturbine (HAWT). Vertical axis wind turbines (VAWT) are a knownalternative to HAWTs, and the applicant's above-referenced U.S.application Ser. No. 15/138,000, entitled “Vertical Axis Lift-DrivenWind Turbine with Force Canceling Blade Configuration,” discusses atlength the advantages, and disadvantages, of heretofore known types oflift-driven VAWTs versus HAWTs.

For a variety of reasons, the Industry has focused more on improvingHAWT technology than on developing VAWTs. As pointed out in referenceapplication Ser. No. 15/138,000, utility-scale VAWT technology is not asmature as that for HAWTs, with no known VAWT systems currently beingoffered or produced by existing utility-scale turbine manufacturers.VAWTs produced in the past that were considered utility-scale at thetime are too small to be considered as such (by a factor of ten or more)by current standards. But the focus has been shifting to VAWTs fordeep-water offshore applications where the turbine will be mounted on afloating platform. One reason is the argument that taking into accountfactors such as operation and maintenance costs, capital investment, andother expenses involved in generating electricity, VAWTs' total cost ofenergy (COE) has the potential to be competitive with, or even lowerthan, COEs for HAWT designs. See, for example, Paquette, Joshua, et al.,“Innovative Offshore Vertical-Axis Wind Turbine Rotor Project,” Proc. ofEuropean Wind Energy Assoc., Copenhagen, Denmark, Apr. 16-19, 2012.

However, to become commercially viable VAWTs will still have to be madeat utility-scale sizes without adversely affecting their potential COEadvantage. This is because according to principles of physics, the sizeof a turbine determines the total power P_(w) it can extract from thewind, expressed by the formula:

P _(w)={(½)ρU ³ A}Cp  (1)

where ρ=density of air, U=wind velocity, A=projected area of the turbine(as defined further below), and Cp is the power coefficient, which is ameasure of turbine efficiency. As seen by this equation, P_(w) can beincreased by making the turbine larger (increasing the value of A in eq.1). But increasing the size (projected area A) of the turbine requiresnot only that the principal turbine parts-blades and struts-be longer,but also that they be made more robust. If all three linear dimensionsof the blades and struts must be increased, then the weight and the costof materials increase as the cube of the size (for example, doublingeach linear dimension would increase cost by a factor of eight), whichsimilarly increases COE. The unique wind turbine described in referenceapplication Ser. No. 15/138,000 represents a significant advance towardmaking larger size VAWT-type turbines commercially viable through adesign that lowers the cost-of-material contribution to COE by reducingthe loads on the larger turbine structure that would otherwise requiremore robust blades and struts.

Even as technical advances make it possible to scale up VAWTs, it isanticipated that the present level of resistance to land-based windfarms will continue to increase. National Wind Watch, Inc.(www.wind-watch.org), cites a long list of organizations around theworld opposed to the use of wind as a renewable energy resource (seealso North American Platform Against Wind Power, www.na-paw.org). Whilesome observers have expressed doubt about the technical efficacy ofusing wind power in any manner, many of the arguments against it relateeither to adverse effects of land-based HAWTs on the environment(unsightly appearance, danger to birds, etc.), or on those living inproximity to them (noise, perceived ambient pressure fluctuations,etc.). The FAA and the U.S. military have also expressed concerns aboutlocating HAWT-based wind farms near aviation sites. “Wind TurbineProjects Run Into Resistance,” New York Times. Aug. 26, 2010 (“In 2009,about 9,000 megawatts of proposed wind projects were abandoned ordelayed because of radar concerns raised by the military and the FederalAviation Administration, according to a member survey by the AmericanWind Energy Association. That is nearly as much as the amount of windcapacity that was actually built in the same year, the trade groupsays.”)(http://www.nytimes.com/2010/08/27/business/energy-environment/27radar.html).

As a result, the interest in developing offshore wind farms of any type,whether they comprise HAWTs or VAWTs, has intensified in recent years.In fact, offshore wind farms have long been a favored approach becauseoceans, seas, and large lakes have plentiful locations where prevailingwinds are more reliably and constantly higher than over land. HAWTs arenot particularly suited to deep-water installations for well-knownreasons, some of which are discussed in reference application Ser. No.15/138,000. Most of these reasons are inherent to the HAWTconfiguration, where the heavy turbine rotor and its associated powergeneration equipment are located hundreds of feet above the surface andthe tower supporting them must remain vertical. Presently, the industryhas not determined a practical way around the necessity of locating anHAWT in water shallow enough to secure its supporting platform in theseabed, which means in almost all cases that offshore HAWT wind farmswill still be in sight of land, at least for the foreseeable future. Tobe out of sight of an observer on land, a 200 m tall wind turbine wouldhave to be roughly 35 miles offshore. Most locations in the large bodiesof water of the world are too deep to justify the increase in COEresulting from the cost of infrastructure required to secure the HAWT tothe underlying seabed.

Another reason for increasing interest in VAWTs is that theirconfiguration makes them much more suitable for floating offshoreinstallation because their low center of gravity makes it easier tostabilize them in the presence of prevailing winds, as discussed in thereferenced application. VAWTs also do not have to be oriented at aspecific angle to the oncoming wind-a significant advantage over HAWTs,which require costly equipment for turning them into the wind. Thedesign approach for free-floating installations of VAWTs has heretoforebeen to seek ways to limit the amount they heel at an angle to verticalin the presence of a prevailing wind. See, for example, Paquette,Joshua, et al., which discusses ways to stabilize floating VAWTplatforms. In Europe, Nenuphar, S. A., of Lille, France, has beenworking since 2006 on bringing floating VAWTs to commercial application.Its approach is to limit the amount by which the VAWT is permitted toheel so that the platform can be made smaller. “Vertiwind: MakingFloating Wind Turbine Technology Competitive for Offshore,” Nenuphar, S.A., October 2012(http://www.twenties-project.eu/system/files/2_2013-03%20Presentation%20short.pdf). More recently, Nenuphar has proposed usingcounter-rotating VAWTs on the same platform, which it claims will reduceheeling moment by at least 40%, and in turn reduce platform cost. “TheNenuphar Solution-Nenuphar Wind,” Nenuphar, S. A., 2015(http://www.nenuphar-wind.com/en/15-the-nenuphar-solution.html). Whilethis approach may work once it is tried on an actual installation, thephysics of using wind energy to generate electricity will probably drivedevelopment toward larger and larger VAWTs to increase their capacity.As VAWTs are made larger, the heeling moment will perforce increase tothe point where platform constructions and gearing arrangements formultiple VAWTs on a single platform may result in unacceptable COEincreases.

What is needed is a fundamentally new design paradigm that will stillenable VAWTs to be scaled up to sizes that can make a meaningfulreduction in their COE (per referenced application Ser. No. 15/138,000),and will also allow offshore installation at distances that place themover the horizon, while avoiding complex and expensive arrangements(like counter-rotating VAWTs mounted on the same platform).

SUMMARY OF THE INVENTION

To that end, a principal aspect of the invention involves approachingthe design of an offshore VAWT-type turbine by taking advantage of thefact that floating VAWTs will necessarily heel in the presence ofprevailing winds, rather than treating heeling as a problem to beovercome. The struts that support the turbine blades of the heeled overVAWT are designed to act as power generating rotors in their own right.This approach uses the increase in total lift in the direction ofrotation created by the blades and the struts to construct a hybridvertical/horizontal axis wind turbine which will assume a predeterminedheel angle at a given wind speed.

In another aspect of the invention, the geometry and aerodynamicproperties of the blades and struts are designed to generate maximumpower at that predetermined heel angle, thus permitting the hybridturbine to be tailored for installation at a particular offshore sitethat has known environmental conditions.

In yet another aspect of the invention, the geometry and the aerodynamicproperties of the blades and struts are chosen to provide essentiallyconstant power over a range of heel angles.

Underlying this innovative design paradigm is the insight that as theheel angle increases, the torque about the hybrid turbine axis generatedby lift on the VAWT blades will decrease, but that the torque generatedby lift on the struts can be made to increase. The invention includes adesign method for determining the optimum heel angle using the complexinteraction between these lift forces for any given turbine geometrywith blades and struts having predetermined aerodynamic characteristicsand the optimum turbine geometry for achieving that heel angle. Byincreasing the torque about the shaft for the wind speed conditions at aspecific location, a power generating system comprising a hybridvertical/horizontal axis wind turbine in accordance with the teachingsherein can be constructed that will generate more power thanconventional VAWTs or HAWTs alone.

Other general and specific aspects, details, embodiments, andadaptations of a VAWT (as defined below) in furtherance of the objectsof the subject matter herein are described below in the context ofcertain specific embodiments of the claimed subject matter.

This Summary is provided solely to introduce in a simplified form aselection of concepts that are described in detail further below. It isnot intended necessarily to identify key or essential features of thesubject claimed herein, nor is it intended to be used an aid indetermining the scope of the claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

The objects of the invention will be better understood from the detaileddescription of its preferred embodiments which follows below, when takenin conjunction with the accompanying drawings, in which like numeralsand letters refer to like features throughout. The following is a briefidentification of the drawing figures used in the accompanying detaileddescription.

FIG. 1 is a schematic depiction of an embodiment of a power generatingsystem with a floating platform mounting a hybrid vertical/horizontalaxis wind turbine adapted from the tilted rotor VAWT in referenced U.S.application Ser. No. 15/138,000.

FIG. 2 is a schematic depiction of the operable geometry of anembodiment of the hybrid wind turbine shown in FIG. 1.

FIG. 3 is a sectional view of the hybrid wind turbine taken along line3-3 in FIG. 1.

FIG. 4 is a sectional view of a strut of the hybrid wind turbine takenalong line 4-4 in FIG. 3.

FIG. 5 illustrates how the defined parameters aspect ratio AR and hybridscaling factor E together influence the value of the turbine axis heelangle β that will yield the maximum power P_(max) generated by a givenhybrid vertical/horizontal axis wind turbine according to an embodimentof the present invention.

FIG. 6 presents the relationship between ε, β, and AR in a slightlydifferent form, in which ε=1 for all cases and AR varies.

FIG. 7 illustrates the relationship between turbine aspect ratio and themaximum power generated by a representative hybrid wind turbineincorporating principles of the present invention, versus the samerelationship for a comparable VAWT.

FIG. 8 illustrates the relationship between heel angle β and aspectratio for the same representative hybrid wind turbine as in FIG. 7.

FIG. 9 illustrates another aspect of the invention whereby a hybrid windturbine can be designed generate to essentially constant power for abroad range of heel angles.

FIG. 10 is a schematic depiction of the hybrid turbine shown in FIGS. 1and 2 in a stowed configuration.

One skilled in the art will readily understand that the drawings are notstrictly to scale, but nevertheless will find them sufficient, whentaken with the detailed descriptions of preferred embodiments thatfollow, to make and use the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The detailed description that follows is intended to provide specificexamples of particular embodiments illustrating various ways ofimplementing the claimed subject matter. It is written to take intoaccount the level of knowledge of one of ordinary skill in the art towhich the claimed subject matter pertains. Accordingly, certain detailsmay be omitted as being unnecessary for enabling such a person torealize the embodiments described herein.

In general, terms used throughout have the ordinary and customarymeaning that would be ascribed to them by one of ordinary skill in theart. However, some of the terms used in the description herein will beexplicitly defined and that definition is meant to apply throughout. Forexample, the term is “substantially” is sometimes used to indicate adegree of similarity of one property or parameter to another. This meansthat the properties or parameters are sufficiently similar in value toachieve the purpose ascribed to them in the context of the descriptionaccompanying the use of the term. Exact equivalence of many propertiesor parameters discussed herein is not possible because of factors suchas engineering tolerances and normal variations in operating conditions,but such deviations from an exact identity still fall within the meaningherein of being “substantially” the same. Likewise, omission of the term“substantially” when equating two such properties or parameters does notimply that they are identical unless the context suggests otherwise.Similar considerations apply to the term “about.” which is sometimesused herein to indicate that the nominal value of a parameter can vary acertain amount as long as it produces the intended effect or result.

Further, when elements are referred to as being “connected” or“coupled,” the elements can be directly connected or coupled together orone or more intervening elements may also be present. In contrast, whenelements are referred to as being “directly connected” or “directlycoupled,” there are no intervening elements present.

It will also be understood that the term “VAWT” is not limited to a windturbine with a vertically oriented shaft axis when in operation. As inreference U.S. application Ser. No. 15/138,000, the term is meant todescribe a lift-driven wind turbine that has blades spaced radially fromthe shaft with a least a portion of each blade along the length thereofcomprising an airfoil for generating on the blade a lift force having acomponent tangential to the direction of rotation with a magnitude onthe blade capable of rotating the shaft in the presence of a prevailingwind. In addition, it should be understood that the term “heel angle”refers to the mean value of the heel angle of the axis of the hybridturbine described herein considered over a predetermined period of time,since in floating operation the heel angle 3 may fluctuate due toenvironmental conditions such as waves and/or varying wind speed, aswell as the unsteady dynamics of the wind turbine itself.

I. VAWT Geometry for Illustrating Principles Underlying the Invention

FIG. 1 is a schematic depiction of an embodiment of a power generatingsystem, based on a hybrid vertical/horizontal axis wind turbine 10, fordeep-water offshore installations. FIG. 1 is adapted from the depictionin FIG. 4 of U.S. application Ser. No. 15/138,000 of a VAWT to whichcertain design principles of the present invention are applicable. Thehybrid turbine 10 includes a rotor 12 that comprises struts 14, 16, 18,and 20, and two blades 22 and 24 mounted by the struts to a towerassembly 26. The blades are designed to cause rotation of a shaftcarried by the tower assembly about an axis A_(R) in the fashion of alift-driven VAWT as known by those skilled in the art and described inthe applicant's referenced U.S. application. The direction of rotationof the rotor is denoted by the arrow drawn around the axis in FIG. 1.This figure illustrates a two-blade rotor, with the blades 22 and 24mounted 180° apart, but the invention contemplates rotors with moreblades, which will preferably be equally spaced circumferentially aroundthe tower.

The hybrid turbine 10 is mounted to an anchored floating supportplatform 30. In this embodiment the platform includes a compartment 32submerged below the water level WL where it houses a conventionalelectrical generator and associated machinery. The platform isillustrated in highly schematic form, and in reality will includesufficient superstructure to support the presence of necessary servicepersonnel. The rotating shaft is operatively connected to the generatorthrough a protective casing 34, which can be made large enough to permitservice personnel to descend to the submerged compartment for repairsand maintenance. Although the generator and associated machinery can beon the platform, the arrangement shown permits the heavy generatingequipment to act as ballast for purposes described further below. Theplatform 30 is moored by cables 38 anchored in the seabed SB to maintainthe turbine 10 at a desired location, but is otherwise free to float topermit the shaft 26 to assume a heel angle !relative to vertical in thepresence of a prevailing wind U_(∞), which in FIG. 1 is from the left inthe plane of the drawing.

The hybrid turbine 10 also incorporates the VAWT force-cancellinggeometry and construction disclosed in reference U.S. application Ser.No. 15/138,000, with the blades 22 and 24 tilted outwardly at an angle γrelative to the shaft and with a geometric angle of attack α_(G)selected as described in that application. Although the presentinvention is advantageously applied to a VAWT with the force-cancelingcharacteristics described in the referenced application, it can beapplied with equal effect to VAWTs (as defined herein) of differentconstructions.

FIG. 2, which is taken from FIG. 7 of the referenced application,illustrates certain dimensional features important to the application ofthe design approach described herein. For ease of description, theturbine is shown in a vertical orientation rather than at a non-zeroheel angle β at which it would operate, and only one strut/bladeassembly is shown for ease of description. The upper struts 14 and 18are hinged at their ends to upper positions of the rotor shaft and therespective blades 22 and 24, while the lower struts 16 and 20 are hingedat their ends to lower positions of the rotor shaft and the respectiveblades 22 and 24. Thus, each strut/blade assembly comprises with therotating shaft a four-bar linkage. The upper strut 14 is connected tothe rotor shaft 40 at an upper shaft hinge represented by the circle 42and to the blade 22 at an upper blade hinge represented by the circle44. The lower strut 16 is connected to the rotor shaft 40 at a lowershaft hinge represented by the circle 46 and to the blade 22 at a lowerblade hinge represented by the circle 48. The hinges 42 and 46 arespaced apart by the distance d, and the hinges 44 and 48 are spacedapart by the distance b. The blade 22 in this embodiment has a length L,as does the blade 24. Both blades are straight, but in other embodimentsthey can have a different geometry.

In an operational configuration depicted in FIG. 2, the struts 14 and 16are rotated from their stowed position (described in more detail inreference application Ser. No. 15/138,000 and below in connection withFIG. 10) to space the blades from the shaft as shown in FIG. 2. Thestruts have different lengths L₁ (strut 16) and L₂(strut 14), whichcause the blade 22 to assume an operational position in which it istilted outwardly at an angle γ with the shaft (see FIG. 1). Theprojected area of the turbine (A in equation no. 1) is twice the shadedarea in FIG. 2. It will be appreciated that A can be changed accordingto the positions of the struts, in accordance with principles discussedin reference application Ser. No. 15/138,000. This feature can be usedto advantage in the present design as described in further detail below.

Another important feature of the turbine rotor 12 is that the struts 14,16, 18, and 20 comprise airfoil sections, which will be understood withreference to FIGS. 3 and 4. FIG. 3 is a section through the turbine 10taken at lines 3-3 in FIG. 1. It depicts a rotor constructed inaccordance with the principles described in U.S. application Ser. No.15/138,000. That is, the mean chord lines C_(b) of the blades 22 and 24have a non-zero geometric angle of attack α_(G), as shown in the figure.(The parameter α_(G) is defined more rigorously in application Ser. No.15/138,000.) However, it will be appreciated that the design principlesaccording to the present invention are equally applicable to prior artVAWTs in which α_(G)=0.

FIG. 4, which is a cross-section taken along line 4-4 in FIG. 3, depictsthe strut 14 or 16 with a strut mean chord line C_(s) which forms apitch angle θ with a reference plane RP perpendicular to the turbineaxis. (θ is positive in the direction shown in the drawing.) Inaccordance with the design principles discussed in detail further below,the pitch angle can be constant along the length of the strut, can vary(twist) as a function of the distance from the turbine axis, or can beadjusted during operation. Similarly, the shape of the strut incross-section and/or the length of its mean chord line C_(s) can beconstant along the length of the strut or can vary as a function of thedistance from the turbine axis A_(R). Those skilled in the art will alsorecognize that there are a variety of ways that the pitch angle can bechanged during operation, such as providing a linkage that rotates thestrut about an axis extending along the strut.

II. Designing a Floating Hybrid Vertical/Horizontal Axis Wind Turbine

The design methodology described herein can be used to construct ahigh-performance, floating hybrid wind turbine. One of the uniqueaspects of the design approach described herein is a relationshipdefined by various turbine parameters that determines a heel angle βthat provides maximum power output for a given turbine construction,which can be used to design a floating hybrid turbine that will assume aheel angle that provides maximum power at a target wind speed, such as aprevailing wind speed at an intended offshore installation site. Thesame design principles can be used to construct a hybrid wind turbinethat generates essentially constant power over a broad range of heelangles.

This design methodology starts with choosing a notional baseline VAWThybrid wind turbine geometry with blades having particular aerodynamicproperties, as if it had no struts. With reference to FIGS. 2 and 3, thebaseline VAWT will have geometric dimensions and aerodynamic propertiesof a VAWT rotor for potential use in the hybrid turbine being designed.The power P_(V0) generated by the baseline VAWT turbine at the designwind speed is calculated using conventional mathematical and numericalmodeling techniques such as computational fluid dynamics (CFD) programs,an example being the CHARM computer software for dynamic modeling offluid flow available from Continuum Dynamics, Inc., of Ewing, N.J.

A similar calculation is done to determine a power P_(H0) for atheoretical baseline HAWT comprising just the struts as if they wererotating about an axis parallel to the wind direction (that is, as anHAWT). In other words, for the turbine shown in FIGS. 1 and 2, each setof struts (such as the struts 14, 18 and struts 16, 20) would be treatedas an HAWT rotor, and the two pairs of struts as two HAWT rotors. Withreference to FIGS. 3 and 4, these baseline HAWT rotors will havegeometric dimensions and aerodynamic properties simulating those of thestruts for potential use in the hybrid turbine being designed. The powerP_(H0) generated by the baseline strut pairs at the target wind speed isalso calculated using conventional mathematical and numerical modelingtechniques.

Then, the following algorithm is used to determine the power contributedby the baseline VAWT of a given size:

$\begin{matrix}{P_{V} \approx {H \times 2 \times R \times \frac{P_{V\; 0}}{A_{V\; 0}} \times \cos \; \beta}} & (2)\end{matrix}$

where H is the height of the turbine, and R is the distance to theblades from the turbine axis (that is, the turbine radius), and A_(V0)is the projected area of the baseline VAWT. Likewise, the followingalgorithm is used to determine the power contributed by struts of agiven size:

$\begin{matrix}{P_{H} \approx {\pi \times R^{2} \times \frac{P_{H\; 0}}{A_{H\; 0}} \times \sin \; \beta}} & (3)\end{matrix}$

where A_(H0) is the projected area of the baseline HAWT (that is, arotor comprising the struts).

Accordingly, the total power P for a hybrid vertical/horizontal axiswind turbine can be approximated by adding equations nos. 2 and 3:

$\begin{matrix}{P \approx {{H \times 2 \times R \times \frac{P_{V\; 0}}{A_{V\; 0}} \times \cos \; \beta} + {\pi \times R^{2} \times \frac{P_{H\; 0}}{A_{{H\; 0}\;}} \times \sin \; \beta}}} & (4)\end{matrix}$

The value of β at which P is a maximum can be determined bydifferentiating equation no. 4 with respect to β, setting the result tozero, and solving for β:

$\begin{matrix}{{\tan \; \beta} = {\left( \frac{P_{H\; 0}}{P_{V\; 0}} \right)\left( \frac{A_{V\; 0}}{A_{H\; 0}} \right) \times \frac{\pi}{2} \times \frac{R}{H}}} & (5)\end{matrix}$

By defining a hybrid scaling factor:

$\begin{matrix}{ɛ = {\left( \frac{P_{H\; 0}}{P_{V\; 0}} \right)\left( \frac{A_{V\; 0}}{A_{H\; 0}} \right)}} & (6)\end{matrix}$

and aspect ratio AR of the hybrid turbine as H/(2R), the followingequation provides the heel angle β that provides maximum power for agiven hybrid turbine aspect ratio:

$\begin{matrix}{{\tan \; \beta} = {ɛ \times \frac{\pi}{4{AR}}}} & (7)\end{matrix}$

It will be appreciated that aspect ratio in this formulation assumesthat the blades 22 and 24 are straight (as noted above) and that R isthe radial distance between the axis A_(R) and the midpoint of the bladefrom either end. In the general case, H is the length L of the bladealong the axis A_(R) (for example, H=L×cos γ for the turbine in FIGS. 1and 2), and R is the mean radial distance between the axis and theblades.

A. Interrelationship of ε, β, and AR

FIG. 5 illustrates how aspect ratio (as defined above) and the parameters together influence the value of β that will yield the maximum powerP_(max) generated by a given hybrid vertical/horizontal axis windturbine. FIG. 5 illustrates that the value of β increases for a givenaspect ratio as ε increases. For example, at AR=2.0, 13 increases from10° at ε=0.5 to 30° at E=1.5. FIG. 6 presents the relationship betweenε, β, and aspect ratio in a slightly different form. In FIG. 6, ε=1 forall of the curves. The line labeled “VAWT” illustrates the manner inwhich the power generated by the VAWT drops as β increases. Thecalculation of the power contributed by the VAWT at different values ofβ was performed using a 5 MW VAWT design (at β=0), and uses the sameVAWT for all values of β. The VAWT aspect ratio changes while theprojected area A_(V0) remains constant (which is why there is only oneVAWT curve in FIG. 6). The contribution to total power by the struts wascalculated for different aspect ratios (AR), but with the value of εkept constant (=1.0). The curves plotted using “open” symbols representthe contribution to power by the struts from zero at β=0, and increasingas β Increases. The curves plotted using “solid” symbols represent thetotal power for the hybrid turbine. To compare FIG. 5 with FIG. 6,consider that the maximum power for AR=1.5 and ε=1.0 is at about β=26°in both FIGS. 5 and 6.

The relationships expressed by the above equations and illustratedvisually in FIGS. 5 and 6 enable a hybrid vertical/horizontal windturbine to be designed to assume an optimum heel angle β at a given sitewith known wind conditions. An example of a design methodology toaccomplish this begins by making certain assumptions regarding theaerodynamic properties of the blades and struts of a baseline VAWT andHAWT, and using known mathematical and/or numerical modeling techniquesto calculate values for P_(V0), A_(V0), P_(H0), and A_(H0), as discussedabove. This provides a value for ε (eq. 6). The turbine will alsotypically have a design power requirement and, in the present example, aspecified wind velocity at which the turbine is to operate. The area ofthe turbine required to generate a given amount of power can becalculated beginning with equation no. 1 above. For purposes of thisexample, it is assumed that Cp is the so-called Betz limit (16/27), sothat the value of A required to generate the target power can beestimated by solving equation no. 1 for A.

At that point, a value of AR can be chosen to provide the heel angle βat which maximum power for the thus-designed turbine will be achieved.Typically, however, there are structural considerations that will affectthe desirability of a turbine with a particular aspect ratio, such asmight be the case for a turbine designed to take advantage of principlesdescribed in reference application Ser. No. 15/138,000. If so, one ormore iterations on the initial design assumptions can be made using theinformation in plots such as FIGS. 5 and 6 to make adjustments to thegeometry of the turbine and/or the aerodynamic characteristics of theblades and struts to find an optimum operating value for the heel angleβ.

It will be appreciated that equation no. 7 is significant for a numberof reasons. First, and fundamentally, it demonstrates that an optimumheel angle β does in fact exist that will generate maximum power for anygiven hybrid wind turbine with the characteristics described above. Butbeyond that, it also provides a basis for actually designing a hybridwind turbine that will assume a predetermined heel angle in the presenceof a prevailing wind at a predetermined velocity. Armed with thesetools, a turbine engineer skilled in the art can design a hybrid turbineto be mounted on a floating platform that will satisfy structural andperformance requirements at a predetermined heel angle for a given windvelocity.

B. Examples of Specific Hybrid Turbine Design Methods

A hybrid vertical/horizontal axis wind turbine according to the aspectsof the invention described above involves first determining theenvironmental conditions at a proposed site and the amount of power tobe produced. From the standpoint of the present design, the mostimportant environmental condition is the prevailing wind speed (thedirection of the prevailing wind being largely irrelevant). Typically,the turbine will be designed for a given wind speed averaged over acertain amount of time. Often this data is already available for aproposed wind farm site, but it can be developed as part of the overalldesign effort.

A more rigorous design method than the one presented above uses striptheory, a known technique whereby the aerodynamics of athree-dimensional blade and strut are modeled as flow over series oftwo-dimensional sections. The following relationship results from usingstrip theory to derive a unique relationship that can be used toestimate the average power over one rotor revolution (360°) generatedthe blades and struts of a hybrid vertical/horizontal axis wind turbineas described herein. This relationship is embodied in the followingequation:

$\begin{matrix}{\frac{\overset{\_}{P}}{\frac{1}{2}\rho \; {U_{\infty}^{3}\left( {2{RL}} \right)}} = {{\frac{N}{2\Lambda^{3}}\frac{c_{blade}}{L}\frac{L}{R}\left\{ {{{\frac{1}{2}\left\lbrack {\frac{\partial C_{L}}{\partial\alpha} - {\frac{3}{2}\left( {C_{d\; 0} + {C_{d\; 2}\alpha_{G}^{3}}} \right)} - C_{d\; 2}} \right\rbrack}\left( {\Lambda \; \cos \; \beta} \right)^{2}} - \left( {C_{d\; 0} + {C_{d\; 2}\alpha_{G}^{2}}} \right)} \right\}_{blade}} + {\frac{M}{2\Lambda^{3}}\frac{c_{strut}}{R}\frac{R}{L}\begin{Bmatrix}{{\frac{\partial C_{L}}{\partial\alpha}\left( {{\frac{1}{3}{\theta\Lambda}\; \sin \; \beta} + {\frac{1}{2}\Lambda^{2}\sin^{2}\beta}} \right)} - {C_{d\; 2}\left( {{\frac{2}{3}\theta \; {\Lambda sin}\; \beta} + {\frac{1}{2}\Lambda^{2}\sin^{2}\beta}} \right)} -} \\{\frac{1}{4}\left( {C_{d\; 0} + {C_{d\; 2}\theta^{2}}} \right)\left( {\Lambda^{2} + 1} \right)}\end{Bmatrix}_{strut}}}} & (8)\end{matrix}$

where “blade” and “strut” indicate the respective contributions of eachto the average power P (defined below), and the terms used in equationno. 8 are as follows:P=power generated by the turbine, averaged over one revolutionβ=heel angleρ=density of airU_(∞)=free stream wind velocityR=the radial distance of the blade from the axis of rotationL=blade lengthN=number of bladesM=number of strutsΩ=angular rotation rate of turbine (in rad/sec)

A=U_(∞)/ΩL

For the blades and struts:c=mean chord length∂C_(L)/∂α=slope of lift coefficient curve with respect to local angle ofattack αC_(d)=drag coefficient, assumed to have the form C_(d)=C_(d0)+C_(d2)×α², where the coefficients are determined by fitting a curve throughexperimental dataC_(d0)=drag coefficient of the airfoil shape at zero angle of attack(α=0)C_(d2)=parameter dependent on airfoil shape used to express thesensitivity of drag coefficient to α²α_(G)=geometric angle of attack of the blades (see FIG. 3)θ=geometric pitch of the struts, defined as the angle from a conicalsurface formed by the strut paths about the turbine axis to a mean chordof the strut airfoil (see FIG. 4)

Equation no. 8 is based on a hybrid turbine in which the blades areparallel to the axis of rotation (A_(R) in FIGS. 1-4) and the struts areperpendicular to the blades and the axis. In addition, geometric angleof attack of the blades and the pitch of the struts are assumed to beconstant along the length L of the blades and length R of the struts.However, the core concepts illustrated by equation no. 8, and the designmethod based on it, apply to more complex turbine geometries and bladeand strut configurations, as well as to turbines in which the blade andstrut configurations and aerodynamic characteristics can be varied.

Equation no. 8 can be simplified and rearranged (and by using thetrigonometric relationship cos ²β=1−sin²β) to provide the followingexpression for the normalized average power per revolution:

$\begin{matrix}{\frac{\overset{\_}{P}}{\frac{1}{2}\rho \; {U_{\infty}^{3}\left( {2{RL}} \right)}} = {C_{0} + {C_{1}\sin \; \beta} + {C_{2}\sin^{2}\beta}}} & (9)\end{matrix}$

Equation no. 9 expresses the average power (normalized with respect toturbine geometry and wind velocity) at a given heel angle β. Thecoefficient C₀ collects the terms in equation no. 8 not multiplied byeither sin β or sin²β; the coefficient C₁ collects the terms in equationno. 8 multiplied by sing; and the coefficient C₂ collects the terms inequation no. 8 multiplied by sin²β. These coefficients comprise thefollowing terms:

$\begin{matrix}{C_{0} = {{\frac{N}{2\Lambda^{3}}\frac{c_{blade}}{L}\frac{L}{R}\left\{ {{{\frac{1}{2}\left\lbrack {\frac{\partial C_{L}}{\partial\alpha} - {\frac{3}{2}\left( {C_{d\; 0} + {C_{d\; 2}\alpha_{G}^{2}}} \right)} - C_{d\; 2}} \right\rbrack}\Lambda^{2}} - \left( {C_{d\; 0} + {C_{d\; 2}\alpha_{G}^{2}}} \right)} \right\}_{blade}} - {\frac{M}{8\Lambda^{3}}\frac{c_{strut}}{R}\frac{R}{L}\left\{ {\left( {C_{d\; 0} + {C_{d\; 2}\theta^{2}}} \right)\left( {\Lambda^{2} + 1} \right)} \right\}_{strut}}}} & \left( {9a} \right) \\{\mspace{20mu} {C_{1} = {\frac{M}{6\Lambda^{2}}\frac{c_{strut}}{R}\frac{R}{L}\left\{ {\left( {\frac{\partial C_{L}}{\partial\alpha} - {2C_{d\; 2}}} \right)\theta} \right\}_{strut}}}} & \left( {9b} \right) \\{C_{2} = {{{- \frac{N}{4\Lambda}}\frac{c_{blade}}{L}\frac{L}{R}\left\{ {\frac{\partial C_{L}}{\partial\alpha} - {\frac{3}{2}\left( {C_{d\; 0} + {C_{d\; 2}\alpha_{G}^{2}}} \right)} - C_{d\; 2}} \right\}_{blade}} + {\frac{M}{4\Lambda}\frac{c_{strut}}{R}\frac{R}{L}\left\{ {\frac{\partial C_{L}}{\partial\alpha} - C_{d\; 2}} \right\}_{strut}}}} & \left( {9c} \right)\end{matrix}$

Differentiating equation no. 9 with respect to β and setting the resultto zero gives the β* that will generate maximum power:

(C ₁+2C ₂ sin β*)cos β*=0  (10)

Since cos β*=0 (β=90°) is not an actual physical solution (it would meanthe turbine is lying on its side), equation no. 10 can be solved for ageneralized design parameter η, which is defined as equal to one formaximum average power P:

$\begin{matrix}{\eta \overset{\Delta}{=}{- \frac{2C_{2}\sin \; \beta}{C_{1}}}} & (11)\end{matrix}$

Thus, when a hybrid vertical/horizontal axis wind turbine is designedwith a predetermined geometry, and blades and struts havingpredetermined aerodynamic characteristics, such that η=1, β=β*, and theturbine will generate maximum power in the presence of a prevailing windat a predetermined velocity. An additional feature is that at windspeeds above the prevailing design value, increased heel angle abovethis optimum would tend to maintain the power generated constant tofully utilize the electric generator's rated output

It will be appreciated that C₂ must be negative, which implies that fora positive heel angle (meaning that the turbine heel in the samedirection as the prevailing wind), C₁ must be positive. If equation no.11 is solved for sing and substituted in equation no. 9, an expressionfor normalized average power p (the left side of equation no. 9) as afunction of Θ:

$\begin{matrix}{{p(\eta)} = {C_{0} + {\frac{C_{1}^{2}}{{4C_{2\;}}\;}\left( {\eta - 2} \right)\eta}}} & (12)\end{matrix}$

For β=0 (vertical turbine axis), η=0 (from equation no. 12), and thenormalized power=C₀. For η=1, the normalized power is maximized (p*) andhas a value:

$\begin{matrix}{p^{*} = {{p(1)} = {C_{0} - \frac{C_{1}^{2}}{4C_{2}}}}} & (13)\end{matrix}$

1. Hybrid Turbine with Optimum Heel Angle

One example of the application of the above design principles to ahybrid turbine as described herein is to determine a heel angle thatmaximizes power generation for a particular turbine geometry and bladeand strut aerodynamics. The turbine geometry will take into account theoverall stresses imposed on the structure as described in referenceapplication Ser. No. 15/138,000. A typical design may have three blades(M=3), each being supported by two struts (N=6). Values for ∂C_(L)/∂α,C_(d0), and C_(d2) above depend on airfoil shape. For convenience, aknown airfoil is usually employed because its precise dimensions andcontour will be available in the literature. A NACA 0012 or NACA 0015airfoil is a typical example of an airfoil suitable for the blades of aVAWT, and can also be used for the struts for convenience. Theseairfoils will yield the following estimated values: ∂C_(L)/∂α=2π,C_(d0)=0.005, and C_(d2)=0.007/(π/18)² (based on curve fits toexperimental data). Those skilled in the art will recognize that 2π willclosely approximate ∂C_(L)/∂∝ for airfoils like NACA 0012 or NACA 0015.See Abbott, I. H., and von Doenhoff, A. E. Theory of Wing Sections:Including a Summary of Airfoil Sections. Dover, 1959, pp. 321, 324(available athttps://aeroknowledge77.flies.wordpress.com/2011/09/58986488-theory-of-wing-sections-including-a-summary-of-airfoil-data.pdf).The mean chord length c for a typical utility-scale VAWT is 1.5 m, whichfor convenience can be used as c for the struts as well. For the sake ofillustration in this example, α_(G)=0 (although it may have a non-zerovalue in a hybrid turbine designed in accordance with referenceapplication Ser. No. 15/138,000), and θ=0.15 rad (≈8.6°).

Assuming U_(∞)/ΩR=1/5, equations nos. 12 and 13 can be used toillustrate the increase in power for different aspect ratios (H/2R,which in the present example is the same as L/2R) for the abovedescribed hypothetical hybrid turbine design. The aspect ratio is chosenas the determining variable for this illustration in accordance with theprinciples described above sections II. and II.A. and the preliminaryscaling law in equation no. 8. According to equation no. 13, the powerfor a VAWT when the axis is vertical (β=0, η=0) is C₀. The maximumaverage normalized power p* (equation no. 13) is greater at all aspectratios, as shown in FIG. 7.

FIG. 8 illustrates how the hypothetical hybrid turbine with the overallgeometry and aerodynamic characteristics is designed for maximum powergeneration for a given aspect ratio. Generally speaking, the aspectratio of a wind turbine configured as a VAWT (as defined above) is animportant parameter for structural reasons, and also influences therotational velocity Ω in the presence of a given prevailing windvelocity U_(∞), which largely determines the value of the parameter A.Thus, the plot in FIG. 8 enables a turbine designer to choose the valueof (that will provide maximum power for a given aspect ratio. Then, fora hybrid turbine with the desired geometry and aerodynamic andstructural characteristics, the heel angle can be achieved by selectinga suitable weight and weight distribution in the platform 30 that willcause the turbine to assume the optimum heel angle β.

It will be understood that the terms “maximum” power and “optimum” heelangle as used herein actually refer to limited ranges of values. Anyfloating turbine system for which there is an “optimum” heel angle or a“maximum” power will still be within the subject matter claimed hereinif it is designed using the principles described herein to generatepower at a level that is substantially increased over that possible witha conventional VAWT of comparable geometry. For example, “maximum power”would cover a range up to about ±10% of a theoretical maximum power,minus realistic/actual system losses (for example, losses attributableto generator efficiency) that could be generated by a given system, and“optimum heel angle” would cover a range up to about ±10° of atheoretical optimum value of β. Using a narrower range, such as 15%, todefine maximum power and optimum heel angle would yield a hybrid turbinewith even more superior performance.

Another way of considering what constitutes an optimum heel angle looksat the power coefficient Cp of the hybrid turbine at a particular designheel angle. Using the design principles discussed herein, the system canbe configured and the blades and struts can be constructed withaerodynamic characteristics for generating lift forces so that the powercoefficient at a design heel angle has a predetermined value that variesless than 0.5% per degree change in heel angle.

2. Minimizing Effect of Heel Angle on Power Generation

In another application of the principles disclosed herein, avertical/horizontal axis hybrid wind turbine can be designed such that,at a given wind speed, the generated power has only a weak dependence onthe heel angle β. There will still be a heel angle at which maximumpower is generated, but the power only decreases by a small amount asthe heel angle varies. This is advantageous for a floating wind turbinewhose heel angle is changing due to changing wind speeds. Using the sameparameters as the example in section B.1. normalized power p is plottedas a function of tilt angle in FIG. 9 for a turbine with an aspect ratioof H/2R=L/2R=1.8, which is a reasonable value for a VAWT component of ahybrid turbine of the type described herein. FIG. 9 illustrates that thepower curve (p vs. β) is relatively flat, thus indicating that powerdoes not vary significantly with heel angle. One approach to designing ahybrid turbine that is relatively insensitive to heel angle would be tostart with the design protocol already descried in this section B.Geometric parameters such as blade length L, strut length R, and theblade and strut chords c_(blade) and c_(strut) are then iterated uponuntil both a desirable maximum power and insensitivity to heel angle (asdefined herein) are attained. In the example plotted in FIG. 9, theaspect ratio of the turbine was varied and the resulting heel angle wascalculated.

It will be understood that the terms “essentially the same,”“substantially the same,” and “does not vary significantly” as appliedto this aspect of the hybrid actually refer to ranges of values. Anyfloating turbine system for which the power generated is described usingthose or similar terms with respect to a range of heel angles will stillbe within the subject matter claimed if it is designed using theprinciples described herein to generate power at a level that issubstantially increased over that possible with a conventional VAWT ofcomparable geometry. For example, this terminology in its various formswould cover a variation up to about ±10% variation in the heel angle inan operating regime of interest. Using a narrower range, such as ±5%,would yield a hybrid turbine with even more superior performance.

The second advantage of this approach is that for VAWT designs meetingcertain geometric, aerodynamic and operating requirements an optimumheel angle that maximizes generated power can also be found. While theserequirements can be complex, an example of one such requirement isinferred from the simple analysis above and the observation thatexistence of a maximum requires C₂<0. From equation no. 9c:

$\begin{matrix}{{4\Lambda \; C_{2}} = {{{- N}\; \frac{c_{blade}}{L}\frac{L}{R}\left\{ {\frac{\partial C_{L}}{\partial\alpha} - C_{d\; 2} - {\frac{3}{2}\left( {C_{d\; 0} + {C_{d\; 2}\alpha_{G}^{2}}} \right)}} \right\}_{blade}} + {M\; \frac{c_{strut}}{R}\frac{R}{L}\left\{ {\frac{\partial C_{L}}{\partial\alpha} - C_{d\; 2}} \right\}_{strut}}}} & (14)\end{matrix}$

Enforcing C₂<0 and rearranging terms in equation no. 14 to solve for theblade and strut areas:

$\begin{matrix}{{\frac{N}{M}\frac{c_{blade}}{c_{strut}}\frac{L}{R}} > \frac{\left\{ {\frac{\partial C_{L}}{\partial\alpha} - C_{d\; 2}} \right\}_{strut}}{\left\{ {\frac{\partial C_{L}}{\partial\alpha} - C_{d\; 2} - {\frac{3}{2}\left( {C_{d\; 0} + {C_{d\; 2}\alpha_{G}^{2}}} \right)}} \right\}_{blade}} \approx 1} & (15)\end{matrix}$

Equation no. 15 can be interpreted as requiring that the total planformarea of the blades (=N× c_(blade)× L) exceed that of the struts (=M×c_(strut)× R) for C₂<0. Thus, if this constraint is imposed on thehybrid wind turbine design parameters, then from equation no. 11 it willbe known that a power maximizing heel angle does exist. The heel angleitself is found by solving equation no. 11.

C. Summary of Design Methods

The above described methods for achieving predetermined performancegoals in a hybrid vertical/horizontal turbine can be adapted fordesigning a hybrid turbine with blades and struts having more complexaerodynamic properties, such as tilted blades and cambered blades (perreference application Ser. No. 15/138,000), cambered struts, and bladesand/or struts with variable twist, variable mean chord length, etc.While not wishing to be limited to any particular way of taking suchproperties into account, one approach could consider the lift generatedby the blades and struts as functions of the angular position of theturbine shaft and the heel angle. This could be accomplished using wellknown mathematical and numerical modelling as well as CFD programs. Theaverage power for complete revolution (equation no. 8) can then beobtained by integrating over one revolution (360°) of the shaft. Theanalysis can be refined further by using more sophisticated aerodynamicsmodels that account for inflow, momentum deficit, stall, wakedistortion, compressibility and viscous effects. These would include,but not necessarily be limited to) lifting lines, vortex latticemethods, panel or boundary element methods, or CFD techniques to solvethe potential flow, Euler, or Navier-Stokes equations. These types ofmodels would involve tradeoffs in flow modeling fidelity andcomputational cost, and a typical approach would use increasinglysophisticated (and computationally expensive) analyses as a particulardesign is refined.

When using numerical methods to calculate averaged loads and power, orincorporating more advanced aerodynamics models, explicit expressionsfor the optimum heel angle β* are often not available. Instead, thisangle is determined numerically using iteration. There are many methodsfor performing such optimization. One option is to compute the averagepower for a sequence of heel angles (for example in the range 0°<β<45°),which permits a determination to be made of the heel angle that producesthe highest power. Similar techniques can be used to also optimizerotation rate, geometry, material and airfoil characteristics, whileensuring that desirable constraints are met such as staying withinmaximum and fatigue stress limits, limiting displacements, operatingaway from stall, etc.

III. Varying the Heel Angle β for Different Environmental Conditions.

Another versatile feature of an embodiment represented by the hybridvertical/horizontal axis wind turbine shown in FIGS. 1 and 2 is that itsgeometry can be varied to adjust the heel angle β for varyingenvironmental conditions, such as seasonal wind speed fluctuations andchanges in sea state (wave height and periodicity, for example). Thiscan be done in a variety of ways, some of which will be described indetail.

A. Changing the Turbine Geometry

As mentioned above, the hybrid vertical/horizontal axis wind turbine 10includes a mechanism for varying its geometry in the fashion describedin application Ser. No. 15/138,000. This is depicted in FIG. 10, whichshows an actuator 100 represented schematically by the numeral 100 inFIG. 2. The actuator is shown with a hydraulic cylinder 102 that moves apiston 104 linearly in the directions indicated by the arrows adjacentto the piston in FIG. 2. The base of the cylinder 102 is connected at ahinge 106 to a slider 108 that moves vertically on the shaft 40. Thedistal end of the piston 104 is connected at a hinge 110 to the lowerstrut 16. As the actuator piston 104 moves in and out of the cylinder,the slider moves to raise and lower the blade 22. FIG. 10 shows theblade 22 in a stowed position discussed in further detail below. Withthe four-bar linkage arrangement of the present embodiment, the motionof the lower strut 16 ensures that the blades 22 and 24 movesimultaneously upward and that all of the struts rotate upward from thepositions shown in FIG. 10.

Not only can the actuator be used to place the turbine in a stowedconfiguration during high winds and other heavy weather, but it can beused to change the geometry of the turbine in accordance with the designprinciples discussed above to change the turbine aspect ratio andprojected area as an aid to maximizing power generation in differentenvironmental conditions.

B. Adjusting Platform Ballast

The infrastructure of the platform 30 and the compartment 32 housing theelectrical generator and associated machinery disposed below thewaterline WL will inherently act as ballast to stabilize thefree-floating system depicted in FIG. 1. The system is designed toassume a target heel angle, per the above discussion, and the weightdistribution of the entire system is taken into account in determiningthe heel angle at which the turbine axis will operate under nominalconditions. However, in some installations the heel angle can be changedto account for different environmental conditions (due to factors suchas seasonal variations in wind speed and/or sea state).

Changing the center of gravity of the system, preferably of the platformand its associated structure, will enable the heel angle to be changedto account for different conditions. For example, adjusting the platformballast can be used to raise or lower the center of gravity (CG) of thefloating platform 30 in FIG. 1. Lowering the compartment 32, or addingmass below an existing CG would increase the righting moment on theplatform and decrease the heel angle β. Conversely, raising the CG wouldmake the platform less stable and increase the heel angle β. Means forchanging the center of gravity could include ballast tanks in whichwater can be displaced by air (or vice versa) or machinery for raisingand lowering the compartment on tracks provided for the purpose.

C. Adjusting Mooring Cable Length

The heel angle β can also be changed by increasing or decreasing thelength of the mooring cables 38 (FIG. 1). For example, if the turbine isheeled at angle β away from the wind, as in FIG. 1, decreasing thelength of an upwind cable or cables (to the left in FIG. 1) whileincreasing the length of a downwind cable or cables will decrease 3.Those skilled in the art will recognize that this can be done in avariety of ways, such as by a winch or windlass carried by the platform

D. Adjusting Platform Counterbalance

The heel angle can also be changed by moving a counterbalance in theupwind (windward) direction so as to provide a righting moment on theplatform and reduce the heel angle β. Conversely, moving thecounterbalance downwind would increase j. Again, those skilled in theart will recognize that this can be in a number of different ways, suchas having an arm extending from the platform with a weight the positionof which can be moved to change the center of gravity of the system.

IV. Summary and Conclusion

Those skilled in the art will readily recognize that only selectedpreferred embodiments of the invention have been depicted and described,and it will be understood that various changes and modifications can bemade other than those specifically mentioned above without departingfrom the spirit and scope of the invention, which is defined solely bythe claims that follow.

1.-11. (canceled)
 12. A method of designing a wind-driven powergenerating system comprising a hybrid wind turbine having a turbinerotor including a shaft rotatable about an axis, at least two rotorblades, and at least two struts, with each of the rotor blades beingmounted to the shaft by at least one of the struts so that at least aportion of each of the rotor blades along the length of said rotor bladeis spaced radially from the shaft, and a floating platform forsupporting the hybrid wind turbine on a body of water for permitting theturbine axis to freely assume an operating heel angle β relative tovertical in the presence of a prevailing wind, the method comprising:setting a target velocity of the prevailing wind; designing the portionof each of the rotor blades spaced radially from the shaft to have apredetermined configuration and aerodynamic characteristics forproducing on the rotor blades a lift force having a component in thedirection of rotor blade rotation at a designated non-zero value of β inthe presence of the prevailing wind at the target velocity; anddesigning each of the struts with at least a portion to have apredetermined configuration and aerodynamic characteristics forproducing on each of the struts a lift force having a component in adirection traveled by said struts as the rotor blades rotate around theshaft axis at the designated non-zero value of β in the presence of theprevailing wind at the target velocity; determining a quantitativerelationship between the geometry of the system and the value of β; andusing the quantitative relationship between the geometry of the systemand the value of β to configure the system so that the turbine shaftheels at the designated non-zero value of β in the presence of theprevailing wind at the target velocity.
 13. The method of designing awind-driven power generating system described in claim 12, furthercomprising designing said portions of said rotor blades and said strutswith configurations and aerodynamic characteristics that producerespective lift force components in the direction of rotor bladerotation so that in the presence of the prevailing wind at the targetvelocity the system generates more electrical power when the turbineshaft heels at the designated non-zero value of β in the presence of theprevailing wind at the target velocity than when β=0.
 14. The method ofdesigning a wind-driven power generating system described in claim 12,further comprising designing the rotor blades and the struts withconfigurations and aerodynamic characteristics that provide an upwardlydirected lift force component as said rotor blades and said strutsrotate in the presence of the prevailing wind at the target velocity forcounteracting the weight of said rotor blades and struts.
 15. The methodof designing a wind-driven power generating system described in claim14, further comprising designing the rotor blades and struts so that thetotal upwardly directed lift force component on the rotor blades andstruts is substantially equal to the weight of said rotor blades andstruts in the presence of the prevailing wind at the target velocity 16.The method of designing a wind-driven power generating system describedin claim 14, further comprising designing each of the rotor blades to besubstantially straight for its entire length and to form a non-zeroangle with vertical when the turbine shaft is substantially vertical.17. The method of designing a wind-driven power generating systemdescribed in claim 12, further comprising: designing the portion of eachof the rotor blades as an airfoil mounted to the struts with saidairfoil at a non-zero geometric angle of attack α_(G) formed from achord line of the airfoil along the portion of said rotor blade to atangent to the rotational path of said rotor blade; and choosing α_(G)so that the magnitude of said lift force radial component on each of theat least two rotor blades has a mean value over one revolution of theturbine shaft that is directed radially inwardly for counteractingcentrifugal force on said rotating rotor blade.
 18. The method ofdesigning a wind-driven power generating system described in claim 17,further comprising configuring each of rotor blades to provide anupwardly directed lift force component for counteracting the weight ofsaid rotating rotor blade.
 19. The method of designing a wind-drivenpower generating system described in claim 18, further comprisingdesigning the rotor blades so that a mean value of the total upwardlydirected lift force component on each of the rotor blades issubstantially equal to the weight of said rotating rotor blade in thepresence of the prevailing wind at the target velocity.
 20. The methodof designing a wind-driven power generating system described in claim18, further comprising spacing a top end of the portion of each of therotor blades farther from the turbine shaft than a bottom end of theportion.
 21. The method of designing a wind-driven power generatingsystem described in claim 20, further comprising designing each of therotor blades to be substantially straight for its entire length and toform a non-zero angle with vertical when the turbine shaft issubstantially vertical.
 22. The method of designing a wind-driven powergenerating system described in claim 12, further comprising configuringeach of the rotor blades to provide an upwardly directed lift forcecomponent so that a mean value of the total upwardly directed lift forcecomponent on each of the rotor blades is substantially equal to theweight of said rotating rotor blade and the struts that mount said rotorblades to the turbine shaft in the presence of the prevailing wind atthe target velocity.
 23. The method of designing a wind-driven powergenerating system described in claim 12, further comprising designingthe portion of each of the at least two struts with predeterminedaerodynamic characteristics as an airfoil with a non-zero pitch angle θdefined as an angle from a conical surface formed by a path each of theat least two struts traces as it travels about the turbine shaft axis toa mean chord of the airfoil.
 24. The method of designing a wind-drivenpower generating system described in claim 12, further comprisingdesigning each of the rotor blades to have a non-straight geometry. 25.A method of generating electrical power using a wind-driven powergenerating system comprising a hybrid wind turbine having a turbinerotor including a shaft rotatable about an axis, at least two rotorblades, and at least two struts, with each of the rotor blades beingmounted to the shaft by at least one of the struts so that at least aportion of each of the rotor blades along the length of said rotor bladeis spaced radially from the shaft, wherein: the hybrid wind turbine ismounted on a floating platform supporting the hybrid wind turbine at apredetermined location on a body of water for permitting the turbineaxis to freely assume an operating heel angle β relative to vertical inthe presence of a prevailing wind, the portion of each of the rotorblades spaced radially from the shaft has a predetermined configurationand aerodynamic characteristics producing a lift force on the rotorblades having a component in the direction of rotor blade rotation at adesignated non-zero value of β in the presence of the prevailing wind,each of the struts with at least a portion has a predeterminedconfiguration and aerodynamic characteristics for producing on each ofthe struts a lift force having a component in a direction traveled bysaid struts as the rotor blades rotate around the shaft axis at thedesignated non-zero value of β in the presence of the prevailing wind,and the system is configured by determining a quantitative relationshipbetween the geometry of the system and the value of β so that theturbine shaft will heel at the designated non-zero value of β in thepresence of the prevailing wind at a target velocity, the methodcomprising: determining the target velocity of the prevailing wind basedon historical environmental data at the predetermined location; anddeploying the system on the body of water at the predetermined location.26. The method of generating electrical power described in claim 25,further comprising configuring the system and designing said portions ofsaid rotor blades and said struts with aerodynamic characteristics sothat in the presence of the prevailing wind at the target velocity thesystem generates more electrical power at the designated non-zero valueof β than at heel angles greater than or less than said value.
 27. Themethod of generating electrical power described in claim 26, furthercomprising configuring the system and designing the rotor blades and thestruts with aerodynamic characteristics so that in the presence ofprevailing winds at velocities different from the target velocity thepower coefficient of the hybrid wind turbine varies less than 0.5% perdegree change in the designated non-zero operating heel angle up tovalues of β of at least 35°.
 28. The method of generating electricalpower described in claim 26, further comprising altering the geometry ofthe system to change the designated non-zero value of β when theprevailing wind has a velocity different from the target velocity. 29.The method of generating electrical power described in claim 28, furthercomprising changing the designated non-zero value of β by altering thegeometry of the turbine rotor.
 30. The method of generating electricalpower described in claim 28, further comprising changing the designatednon-zero value of β by altering the floating platform to change thedistribution of weight in the floating platform.
 31. The method ofmethod of generating electrical power described in claim 25, comprisingconfiguring the system so that in the presence of prevailing winds atvelocities different from the target velocity the power coefficient ofthe hybrid wind turbine varies less than 0.5% per degree change in thedesignated non-zero operating heel angle up to values of the operatingheel angle of at least 35°.
 32. The method of generating electricalpower described in claim 25, wherein: the struts comprise an upper strutand a lower strut pivotally mounted to the turbine shaft and associatedrotor blade with the upper strut mounted to the shaft and the associatedrotor blade at respective vertical heights greater than the lower strut;the upper struts are substantially identical and the lower struts aresubstantially identical, with the upper struts being longer than thelower struts so that the top of the portion of each of the at least tworotor blade is farther from the turbine shaft than the bottom of saidportion; each of the rotor blades and the struts associated with saidrotor blade comprise a four-bar linkage permitting radial movement ofsaid rotor blade relative to the turbine shaft between a rotor operativeposition for rotation by the prevailing wind to drive the electricalgenerator and a rotor stowed position in which the rotor blades areproximate to the turbine shaft for protection from excessive prevailingwinds, the method further comprising: designing the rotor blades and thestruts with configurations and aerodynamic characteristics that generatelift forces providing an upwardly directed lift force component thatcounteracts the weight of the rotating rotor blades and the struts andurges the rotor blades into the operative position in the presence ofthe prevailing wind.
 33. The method of generating electrical powerdescribed in claim 32, further comprising moving the rotor bladesbetween the operative position and the stowed position using an actuatoroperatively connected to the rotor.
 34. The method of generatingelectrical power described in claim 33, further comprising using theactuator to hold the at least two rotor blades at a predetermined radialposition relative to the shaft.
 35. The method of generating electricalpower described in claim 34, further comprising designing the rotorblades with configurations and aerodynamic characteristics so that amean value of the total upwardly directed lift force component on eachof the rotor blades is substantially equal to the weight of the rotatingrotor blade and the struts associated with said rotor blade formaintaining the rotor blades in the operative position in the presenceof the prevailing wind.
 36. The method of generating electrical powerdescribed in claim 25, wherein each of the rotor blades has anon-straight geometry.